3.4 Verification of Analytical Model
3.4.4 Mutual Inductance
The proposed analytical model is used to calculate the mutual inductance for different misalign- ments. The mutual inductance from the analytical model is calculated by surface integration of the magnetic field in Region V obtained from (3.12), over the rectangular surface of dimension 2a2 and 2b2. The mutual inductance M is expressed as follows:
M = 1 Ip
ZZ
BV.dA (3.26)
where Ipis the rms current flowing in primary coil.
TH-2341_126102029
3. 3-D Analytical Model for Computation of Mutual Inductance for Different Misalignment with Shielding in WPT System
The comparison of variations in mutual inductance without shielding and with shielding, obtained from analytical, FEA, and measurement are shown in Figs. 3.13-3.18. In the Figs. 3.13-3.18, for each misalignment, variation in mutual inductance with shielding shows the same pattern with a higher magnitude compared to without shielding. This is due to the reduction in leakage flux.
1 2 3 4 5 6 7 8 9 10 11 12
Vertical distance (cm) 0
0.5 1 1.5 2 2.5 3
Mutual inductance (H)
×10-5
3D Analytical 3D FEM Experimental µr=1
(a)
1 2 3 4 5 6 7 8 9 10 11 12
Vertical distance (cm) 0
0.5 1 1.5 2 2.5 3
Mutual inductance (H)
×10-5
3D Analytical 3D FEM Experimental µr=2600
(b)
Figure 3.13: Mutual inductance variation in z-direction (a) without shielding and (b) with shielding.
Fig. 3.13 shows the variation of mutual inductance as the secondary coil moves along the z- direction. Here, mutual inductance decreases as the secondary coil moves along the z-axis because of the decrease in flux linkage.
0 10 20 30 40 50 60 70 80 90
Angle (° ) 0
1 2 3 4
Mutual inductance (H)
×10-6
3D Analytical 3D FEM Experimental
µr=1
(a)
0 10 20 30 40 50 60 70 80 90
Angle (° ) 0
1 2 3 4
Mutual inductance (H)
×10-6
3D Analytical 3D FEM Experimental
µr=2600
(b)
Figure 3.14: Mutual inductance variation for angular misalignment (a) without shielding and (b) with shielding.
The variation in mutual inductance due to angular misalignment is shown in Fig. 3.14. For the measurement, the centre position of the secondary coil is kept at z = 10.25 cm, and the coil is tilted around the y-axis from 0◦ to 90◦ with an interval of 10◦. Here, until 40◦, both the cease small variation in mutual inductance is observed, and after this, it decreases drastically. In this variation, Bx and Bzboth contribute to mutual inductance as the coil tilt the contribution of Bzin mutual inductance TH-2341_126102029
3.4 Verification of Analytical Model
decreases.
0 20 40 60 80 100 120 140 160 180
Angle (° ) 0
1 2 3 4
Mutual inductance (H)
×10-6
3D Analytical 3D FEM Experimental µr=1
(a)
0 20 40 60 80 100 120 140 160 180
Angle (° ) 0
1 2 3 4
Mutual inductance (H)
×10-6
3D Analytical 3D FEM Experimental µr=2600
(b)
Figure 3.15: Mutual inductance variation for planar misalignment at z = 10.25 cm (a) without shielding and (b) with shielding.
Fig. 3.15 shows the variation in mutual inductance for a planar misalignment. Since in this variation, the plane of the secondary coil is in parallel with the plane of the primary coil. Therefore only Bzcontributes to mutual inductance. Here, it is observed that mutual inductance decreases for 0◦ to 90◦rotation of the secondary coil and again increases for 90◦ to 180◦rotation. This is due to a decrease in the overlapping area between coils during the rotation of 0◦to 90◦and an increase in the overlapping area during the rotation of 90◦to 180◦.
The variation in mutual inductance for the case where secondary coil moves along the x -direction are reported in Figs. 3.16-3.18. It may be seen from Figs. 3.16-3.18, as the secondary coil moves away from the centre of the primary coil, there is a decrease in the overlapping area between the coils, and hence, the mutual inductance between the coils gets reduced.
0 2 4 6 8 10 12
Horizontal distance (cm) 0
0.5 1 1.5 2 2.5 3
Mutual inductance (H)
×10-5
3D Analytical 3D FEM Experimental
µr=1
(a)
0 2 4 6 8 10 12
Horizontal distance (cm) 0
0.5 1 1.5 2 2.5 3
Mutual inductance (H)
×10-5
3D Analytical 3D FEM Experimental
µr=2600
(b)
Figure 3.16: Mutual inductance variation for horizontal misalignment at z = 1.35 cm. (a) without shielding and (b) with shielding.
TH-2341_126102029
3. 3-D Analytical Model for Computation of Mutual Inductance for Different Misalignment with Shielding in WPT System
In Fig. 3.17 angular with horizontal misalignment (AHM) is illustrated. Here, the secondary coil moves along x-direction with the angular rotation of 10◦and 20◦around the y-axis, having its centre at z= 3.2 cm and 4.74 cm, respectively. From Fig. 3.17 it can be seen that horizontal misalignment for 10◦rotation has a higher magnitude mutual inductance as compare to 10◦rotation.
For the measurement of variation in mutual inductance due to planar with horizontal misalignment (PHM) is shown in Fig. 3.18. Here, the secondary coil moves along x-direction having z = 2.4 cm and planar rotation of 20◦ (around z-axis). Among all the misalignments shown in Figs. 3.13-3.18, it can be observed that planar misalignment shows the small variation in mutual inductance during the misalignment.
0 2 4 6 8 10 12
Horizontal distance (cm) 0
0.5 1 1.5 2
Mutual inductance (H)
×10-5
3D Analytical 3D FEM Experimental µr=1
10°
20°
(a)
0 2 4 6 8 10 12
Horizontal distance (cm) 0
0.5 1 1.5 2
Mutual inductance (H)
×10-5
3D Analytical 3D FEM Expeimental
20°
10° µr=2600
(b)
Figure 3.17: Mutual inductance variation for both angular and horizontal misalignment. (a) without shielding and (b) with shielding.
0 2 4 6 8 10 12
Horizontal distance (cm) 0
0.5 1 1.5 2
Mutual inductance (H)
×10-5
3D Analytical 3D FEM Experimental
µr=1
(a)
0 2 4 6 8 10 12
Horizontal distance (cm) 0
0.5 1 1.5 2
Mutual inductance (H)
×10-5
3D Analytical 3D FEM Experimental
µr=2600
(b)
Figure 3.18: Mutual inductance variation for both planar and horizontal misalignment. (a) without shielding and (b) with shielding.
The magnetic field obtained in Region IV is used to calculate the self-inductance (Lp) of the TH-2341_126102029
3.4 Verification of Analytical Model
primary coil (with and without shielding). The expression for the self-inductance is given in (3.27) Lp = 1
Ip
ZZ
BIV.dA (3.27)
The obtained self-inductances (with and without shielding) from an analytical model are compared in Table 3.2 with measured values. The self-inductances are measured at 100 kHz with the help of LCR meter.
Table 3.2: Comparison of the primary self-inductance (Lp) Coil Analytical (µH) Measured (µH)
Without shielding 28.41 28.74
With shielding 41.12 40.50
In this subsection, the variation in mutual inductance for different misalignments are shown in Figs. 3.13-3.18. It is observed that for the coil system shown in Fig. 3.1, mutual inductance variation obtained from the analytical model, FEA, and experiment are following the same pattern, which confirms the accuracy of the proposed analytical model. The next subsection compares proposed analytical results with the analytical model reported in [1].